They allow strollers on all of the buses, but you will need to consider your timing carefully:

The bus that services the Hilton directly (without having to walk to Downtown Disney) will be very crowded by the time it reaches the Hilton, as it stops at other hotels along that strip first. You will most likely need to stand when going *to* the park. On your return trips, when taking that bus, it is not uncommon to have to wait upwards of 45 minutes or so to get a bus at park closing, as the lines get long fast, and you might choose to wait for the next bus so you can sit down rather than cram into a standing room only one.

Walking to Downtown Disney is your other option, but you will also run into the line problem. During the day, this is generally not an issue, but after parades and at park closures, the lines for all of the buses (and monorails, and boat launches) get really long and you'll have to wait. With the boats and monorails, you have priority with the strollers, but with the buses you're stuck in line with everyone else, so, again, you might choose to let a bus go, even after waiting 45 minutes already, just so you can sit down.

In my many years of experience at Disney World I've come up with the theory that the buses to people is inversely proportional to the amount of money spent on their room. For example: The Yacht, Beach, and Boardwalk all cost upwards of $300 a night, yet buses show up more often (and therefore there are never lines) for them than for the Pop and All Star Resorts ($100 a night), which have lines running all the way back to the park.

The Hilton is on Resort Row, where the rooms can be Pricelined (so can the Swan and Dolphin, but rarely) for as low as $30 a night, so all those hotels share one bus at times. That line actually does move faster than the one for the All Stars, but its a 15 minute + ride from the Magic Kingdom to the Hilton, so if you're motion sick at all definitely be prepared to wait for a bus with seats.

Personally, I always stay in an Epcot resort. In years where I don't want to spend too much money, I stay in the Swan or Dolphin, or rent DVC points for the Boardwalk from a DVC member (PM me if you want info on that). You're a boat launch to Epcot and MGM, and you can take a boat to Epcot and walk to the Monorail (a nice, pretty 1/3 mile or so walk) or just hop on one of the aforementioned nonbusy "expensive resort" buses to the Magic Kingdom or Animal Kingdom.

You can get Starwood Points from the Swan/Dolphin, turn them into AA miles and then back into HH points